Finite Group Databasebeta

← #194031

Subgroups — #194031

Order 512.

1564 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3039C211yesmaximal, characteristic
2#34183C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic, complemented
4C2×#1990C211yesmaximal, characteristic, complemented
5C2×#1476C211yesmaximal, characteristic
6#8987C214yesmaximal, complemented
7#8985C214yesmaximal, complemented
8#8043C211yesmaximal, characteristic, complemented
9#7897C211yesmaximal, characteristic, complemented
10C22×#137—21no—
11C22×C4⋊Q8C2211yescharacteristic
12C2×#520—21no—
13C2×C4×#41C2211yescharacteristic
14#3039—24no—
15#3039C418yescomplemented
16C2×#462—21no—
17C2×#462C2212yes—
18C2×#461—21no—
19C2×#461C2212yes—
20C2×#431C2211yescharacteristic
21C2×#423C2211yescharacteristic
22C2×#419—21no—
23C22×C4×C8C2211yescharacteristic
24Q8×C4⋊C4—21no—
25Q8×C4⋊C4C2212yes—
26D4×C4⋊C4—22no—
27D4×C4⋊C4—21no—
28D4×C4⋊C4—22no—
29D4×C4⋊C4C2212yes—
30#2317—22no—
31#2317—21no—
32#2285—21no—
33#2285C2211yescharacteristic
34C2×#384—21no—
35C2×#384C2211yescharacteristic
36#1990—22no—
37#1990—21no—
38#1990—21no—
39#1990C2214yes—
40#1988C2214yes—
41#1823—22no—
42C2×#340—22no—
43#1476—24no—
44C2×Q8⋊C8C2212yes—
45C2×D4⋊C8C2212yes—
46#1296C2214yes—
47#1279C2214yes—
48C22×#41D411yescharacteristic
49C23×Q8—21no—
50C2×#137—28no—
51C2×#137—44no—
52C2×#137—44no—
53C22×Q16—22no—
54C22×SD16—22no—
55C22×M16—21no—
56C23×C8—21no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8C2×C414yescomplemented
64C2×C4⋊Q8C2312yes—
65#520—44no—
66#520—44no—
67C4×#41—22no—
68C4×#41—22no—
69C4×#41C2×C414yes—
70C4×#41D414yes—
71C2×C4×Q8—22no—
72C2×C4×Q8D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—21no—
75C2×C4×D4—21no—
76C2×C4×D4—24no—
77C2×C4×D4—22no—
78C2×C4×D4D411yescharacteristic
79C2×C4×D4C2311yescharacteristic
80C2×#117—22no—
81C2×#117—21no—
82C2×#117—21no—
83C2×#117D411yescharacteristic
84C22×C4⋊C4—21no—
85C22×C4⋊C4—21no—
86C22×C4⋊C4—22no—
87C22×C4⋊C4—21no—
88C22×C4⋊C4—22no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—22no—
91C22×C4⋊C4D412yes—
92C22×C42—21no—
93C22×C42C2311yescharacteristic
94#462—28no—
95#462—24no—
96#462C2×C418yescomplemented
97#461—28no—
98#461—24no—
99#461C2×C418yescomplemented
100#431—22no—
101#431—22no—
102#423—22no—
103#423C2×C414yescomplemented
104C2×C4⋊C8—21no—
105C2×C4⋊C8C2312yes—
106C2×C4⋊C8C2312yes—
107#419—28no—
108C2×#103—22no—
109C2×#103—22no—
110C2×C22⋊C8—21no—
111C2×C4×C8—22no—
112C2×C4×C8—22no—
113C2×C4×C8—21no—
114C2×C4×C8—21no—
115C2×C4×C8C2312yes—
116#389—24no—
117#389—22no—
118#389—24no—
119#389—22no—
120#385—24no—
121#385—24no—
122#385—24no—
123#384—22no—
124#384—22no—
125#384—22no—
126#384—21no—
127#384—21no—
128#384D412yes—
129#384D411yescharacteristic
130#384D411yescharacteristic
131#384C2312yes—
132#382—22no—
133#382—21no—
134#380—22no—
135#380—21no—
136#380—22no—
137#380D412yes—
138C4×C4⋊C4—22no—
139C4×C4⋊C4—22no—
140C4×C4⋊C4—21no—
141C4×C4⋊C4C2312yes—
142#340—24no—
143#340—24no—
144#340—28no—
145Q8⋊C8—24no—
146D4⋊C8—24no—
147C2×#41—22no—
148C2×#41—22no—
149C2×#41—22no—
150C2×#41C22⋊C414yes—
151C2×#41D814yes—
152C2×#41SD1614yes—
153C22×Q8—24no—
154C22×Q8—22no—
155C22×Q8—41no—
156C22×Q8—42no—
157C22×Q8—42no—
158C22×Q8—42no—
159C22×Q8—42no—
160C22×Q8—42no—
161C22×Q8C2×D411yescharacteristic
162C22×D4—21no—
163C22×D4C2×D411yescharacteristic
164C23×C4—21no—
165C23×C4—21no—
166C23×C4—21no—
167C23×C4—24no—
168C23×C4—22no—
169C23×C4—41no—
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173#137—432no—
174C2×Q16—28no—
175C2×Q16—44no—
176C2×Q16—44no—
177C2×SD16—28no—
178C2×SD16—44no—
179C2×SD16—44no—
180C2×M16—24no—
181C2×M16—42no—
182C2×M16—42no—
183C22×C8—24no—
184C22×C8—22no—
185C22×C8—22no—
186C22×C8—22no—
187C22×C8—42no—
188C22×C8—42no—
189C4⋊Q8—24no—
190C4⋊Q8—24no—
191C4⋊Q8—24no—
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—22no—
195C4⋊Q8—42no—
196C4⋊Q8—42no—
197C4⋊Q8—42no—
198C4⋊Q8C22×C414yes—
199C4×Q8—22no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—44no—
203C4×Q8—44no—
204C4×Q8C22⋊C414yes—
205C4×Q8C22×C412yes—
206C4×Q8C2×D412yes—
207C4×D4—24no—
208C4×D4—24no—
209C4×D4—24no—
210C4×D4—24no—
211C4×D4—22no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—416no—
215C4×D4—44no—
216C4×D4—44no—
217C4×D4C22⋊C414yes—
218C4×D4C22×C412yes—
219C4×D4C2×D412yes—
220#117—22no—
221#117—24no—
222#117—24no—
223#117—48no—
224#117—42no—
225#117—42no—
226#117C22⋊C414yes—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—24no—
230C2×C4⋊C4—24no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—24no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—22no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—22no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4—48no—
246C2×C4⋊C4—41no—
247C2×C4⋊C4—41no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4—42no—
250C2×C4⋊C4—42no—
251C2×C4⋊C4—42no—
252C2×C4⋊C4—44no—
253C2×C4⋊C4—84no—
254C2×C4⋊C4—82no—
255C2×C4⋊C4—84no—
256C2×C4⋊C4C2×D412yes—
257C2×C4⋊C4C2×D412yes—
258C2×C4⋊C4C2×D411yescharacteristic
259C2×C4⋊C4C2×D411yescharacteristic
260C2×C4⋊C4C2×D414yes—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C22⋊C4—24no—
265C2×C22⋊C4—22no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—24no—
270C2×C42—24no—
271C2×C42—24no—
272C2×C42—22no—
273C2×C42C22×C412yes—
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D412yes—
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42#4112yes—
279C2×C42#4112yes—
280C2×C42C2411yescharacteristic
281C4⋊C8—24no—
282C4⋊C8—22no—
283C4⋊C8—22no—
284C4⋊C8—22no—
285C4⋊C8C22×C414yes—
286#103—28no—
287#103—28no—
288C22⋊C8—42no—
289C4×C8—22no—
290C4×C8—22no—
291C4×C8—42no—
292C4×C8—42no—
293#95—24no—
294#95—24no—
295#95—22no—
296#95—22no—
297C24—21no—
298C24#12011yescharacteristic
299#41—28no—
300#41—28no—
301#41—28no—
302#41#102116yes—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—24no—
307C2×Q8—42no—
308C2×Q8—44no—
309C2×Q8—42no—
310C2×Q8—42no—
311C2×Q8—42no—
312C2×Q8—42no—
313C2×Q8—48no—
314C2×Q8—48no—
315C2×Q8—82no—
316C2×Q8—82no—
317C2×Q8—84no—
318C2×Q8—84no—
319C2×Q8C2×C22⋊C412yes—
320C2×Q8C2×D812yes—
321C2×Q8C2×SD1612yes—
322C2×D4—24no—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—21no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4C2×C22⋊C412yes—
329C2×D4C2×D812yes—
330C2×D4C2×SD1612yes—
331C22×C4—21no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—24no—
335C22×C4—24no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—24no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—24no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—44no—
353C22×C4—48no—
354C22×C4—44no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—84no—
358C22×C4—84no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4C2×C22⋊C412yes—
362C22×C4C4×D412yes—
363C22×C4C4×D412yes—
364C22×C4#12011yescharacteristic
365C22×C4#12011yescharacteristic
366C22×C4#12011yescharacteristic
367C22×C4#12112yes—
368C22×C4#12112yes—
369C22×C4#12312yes—
370C22×C4C2×D812yes—
371C22×C4C2×SD1612yes—
372C22×C4AGL(1,ℤ/8)12yes—
373C22×C4#13712yes—
374C22×C4C22×D411yescharacteristic
375C22×C4C22×D411yescharacteristic
376C22×C4C22×D411yescharacteristic
377Q16—416no—
378SD16—416no—
379M16—48no—
380C2×C8—24no—
381C2×C8—24no—
382C2×C8—42no—
383C2×C8—48no—
384C2×C8—42no—
385C2×C8—42no—
386C2×C8—42no—
387C2×C8—42no—
388C2×C8—42no—
389C2×C8—42no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—24no—
397C4⋊C4—24no—
398C4⋊C4—24no—
399C4⋊C4—44no—
400C4⋊C4—48no—
401C4⋊C4—42no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—44no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—42no—
413C4⋊C4—42no—
414C4⋊C4—82no—
415C4⋊C4—84no—
416C4⋊C4—84no—
417C4⋊C4C2×C22⋊C412yes—
418C4⋊C4C2×C22⋊C412yes—
419C22⋊C4—24no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—24no—
423C22⋊C4—48no—
424C22⋊C4—44no—
425C42—22no—
426C42—22no—
427C42—24no—
428C42—24no—
429C42—48no—
430C42—42no—
431C42—42no—
432C42C2×C22⋊C412yes—
433C42C4×D412yes—
434C42C4×D412yes—
435C42#12112yes—
436C42#12112yes—
437C42#12312yes—
438C42C23×C411yescharacteristic
439C42C22×D411yescharacteristic
440C42C2×#4111yescharacteristic
441C42C2×#4111yescharacteristic
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—22no—
447C23—22no—
448C23—22no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23#38612yes—
454C23#44712yes—
455C23#44812yes—
456C23C2×#12011yescharacteristic
457Q8—22no—
458Q8—22no—
459Q8—22no—
460Q8—44no—
461Q8—44no—
462Q8—82no—
463Q8—82no—
464Q8—88no—
465Q8C2×#10214yes—
466D4—216no—
467D4—22no—
468D4—22no—
469D4—22no—
470D4C2×#10214yes—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—22no—
478C2×C4—216no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—48no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—42no—
493C2×C4—44no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—44no—
497C2×C4—42no—
498C2×C4—44no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—44no—
502C2×C4—88no—
503C2×C4—82no—
504C2×C4—82no—
505C2×C4—164no—
506C2×C4—168no—
507C2×C4#38612yes—
508C2×C4#38612yes—
509C2×C4#38612yes—
510C2×C4C2×#10214yes—
511C2×C4#41714yes—
512C2×C4#44712yes—
513C2×C4#44812yes—
514C2×C4#45512yes—
515C2×C4#45612yes—
516C2×C4#48014yes—
517C2×C4#48114yes—
518C2×C4C22×C22⋊C411yescharacteristic
519C2×C4C2×C4×D411yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C2×#12111yescharacteristic
522C2×C4C2×#12111yescharacteristic
523C2×C4C2×#12311yescharacteristic
524C2×C4C22×D811yescharacteristic
525C2×C4C22×SD1611yescharacteristic
526C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
527C2×C4C2×#13711yescharacteristic
528C2×C4#57712yes—
529C2×C4#57812yes—
530C8—44no—
531C8—44no—
532C8—82no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—24no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22#187414yes—
547C22C2×#38611yescharacteristic
548C22C2×#44711yescharacteristic
549C22C2×#44811yescharacteristic
550C22#300911yescharacteristic
551C22#301111yescharacteristic
552C22C2×#45511yescharacteristic
553C22C2×#45611yescharacteristic
554C4—24no—
555C4—22no—
556C4—21no—
557C4—21no—
558C4—42no—
559C4—42no—
560C4—84no—
561C4—82no—
562C4—84no—
563C4—82no—
564C4#187414yes—
565C4#188214yes—
566C4C22×#10211yescharacteristic
567C4C2×#41711yescharacteristic
568C4#294712yes—
569C4#304612yes—
570C4C2×#48011yescharacteristic
571C4C2×#48111yescharacteristic
572C2—24no—
573C2—44no—
574C2#1863011yescharacteristic
575C2#1863211yescharacteristic
576C2#1865011yescharacteristic
577C2#1865211yescharacteristic
578C2C2×#187411yescharacteristic
579C2#2093111yescharacteristic
580C2C2×#188211yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #21833C2semidirect
4. C2×#1990C2semidirect
6. #8987C2semidirect
7. #8985C2semidirect
8. #8043C2semidirect
9. #7897C2semidirect
15. #3039C4semidirect
63. C2×C4⋊Q8C2×C4semidirect
96. #462C2×C4semidirect
99. #461C2×C4semidirect
103. #423C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#30392
#341832
#218332
C2×#19902
C2×#14762
#89872
#89852
#80432
#78972